Recent Developments in Three-Mode Factor Analysis: Constrained Three-Mode Factor Analysis and Core Rotations

A review is presented of some recent developments in three-mode factor analysis, that are all aimed at reducing the difficulties in interpreting three-mode factor analysis solutions. First, variants of three-mode factor analysis with zero constraints on the core are described, and attention is paid to algorithms for fitting these models, as well as to uniqueness of the representations. Next, various methods for rotation of the core to simple structure are discussed and related to two-way simple structure rotation techniques. In the concluding section, new perspectives for simplification of the interpretation of three-mode factor analysis solutions are discussed.

[1]  Richard A. Harshman,et al.  Foundations of the PARAFAC procedure: Models and conditions for an "explanatory" multi-model factor analysis , 1970 .

[2]  J. Chang,et al.  Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition , 1970 .

[3]  Pieter M. Kroonenberg,et al.  Three-mode principal component analysis : theory and applications , 1983 .

[4]  H. Kaiser.  The varimax criterion for analytic rotation in factor analysis , 1958 .

[5]  R. Henrion.  Body diagonalization of core matrices in three‐way principal components analysis: Theoretical bounds and simulation , 1993 .

[6]  H. Kiers.  Three-way SIMPLIMAX for oblique rotation of the three-mode factor analysis core to simple structure , 1998 .

[7]  G. A. Ferguson,et al.  The concept of parsimony in factor analysis , 1954 .

[8]  J. Leeuw,et al.  Principal component analysis of three-mode data by means of alternating least squares algorithms , 1980 .

[9]  H. Kiers.  Simplimax: Oblique rotation to an optimal target with simple structure , 1994 .

[10]  Henk A. L. Kiers,et al.  Hierarchical relations among three-way methods , 1991 .

[11]  J. Carroll.  An analytical solution for approximating simple structure in factor analysis , 1953 .

[12]  H. Kiers.  Three-mode orthomax rotation , 1997 .

[13]  J. O. Neuhaus,et al.  THE QUARTIMAX METHOD , 1954 .

[14]  David R. Saunders,et al.  AN ANALYTIC METHOD FOR ROTATION TO ORTHOGONAL SIMPLE STRUCTURE , 1953 .

[15]  H. Kiers.  TUCKALS core rotations and constrained TUCKALS modelling , 1992 .

[16]  J. Leeuw,et al.  A new computational method to fit the weighted euclidean distance model , 1978 .

[17]  R. Jennrich.  Orthogonal rotation algorithms , 1970 .

[18]  A. Agresti,et al.  Multiway Data Analysis , 1989 .

[19]  Takashi Murakami.  Quasi Three-Mode Principal Component Analysis — A Method for Assessing the Factor Change — , 1983 .

[20]  Three-mode factor analysis with binary core and orthonormality constraints , 1992 .

[21]  L. Tucker,et al.  Some mathematical notes on three-mode factor analysis , 1966, Psychometrika.

[22]  H. Kaiser,et al.  Oblique factor analytic solutions by orthogonal transformations , 1964 .

[23]  R. MacCallum,et al.  Transformation of a three-mode multidimensional scaling solution to indscal form , 1976 .

[24]  J. Berge,et al.  Uniqueness of three-mode factor models with sparse cores: The 3 × 3 × 3 case , 1997 .

[25]  Pieter M. Kroonenberg,et al.  The TUCKALS line: a suite of programs for three-way data analysis , 1994 .